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Chapter 5: Q 45. (page 429)

Solve the integral: $鈭� {(x - e^{x})}^{2} d x$

Short Answer

Expert verified

The required answer is $\frac{x^{3}}{3} - 2 x e^{x} + 2 e^{x} + \frac{1}{2} e^{2 x} + c$ .

Step by step solution

01

Step 1. Given information.

We have given integral is $鈭� {(x - e^{x})}^{2} d x$ .

02

Step 2. Solve the integration by parts .

We have,

$鈭� {(x - e^{x})}^{2} d x = 鈭� (x^{2} - 2 x e^{x} + e^{2 x}) d x = \frac{x^{3}}{3} - (2 鈭� x e^{x} d x) + 鈭� e^{2 x} d x$

03

Step 3. Integration by parts.

We have,

$u = x d u = d x$

and

$d v = e^{x} d x v = 鈭� e^{x} d x v = e^{x}$

The formula of integration by parts is $鈭� u d v = u v - 鈭� v d u$
$\frac{x^{3}}{3} - (2 (x e^{x} - 鈭� e^{x} d x)) + 鈭� e^{2 x} d x = \frac{x^{3}}{3} - 2 x e^{x} + 2 e^{x} + 鈭� e^{2 x} d x = \frac{x^{3}}{3} - 2 x e^{x} + 2 e^{x} + (\frac{1}{2} 鈭� e^{2 x} d x) = \frac{x^{3}}{3} - 2 x e^{x} + 2 e^{x} + \frac{1}{2} e^{2 x} + c$

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Most popular questions from this chapter

Solve given definite integral.

${鈭�}_{鈭� 1}^{1} 鈥� x^{3} \sqrt{9 x^{2} 鈭� 1} d x$

Solve the integral: $鈭� 3 x e^{x^{2}} d x$

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving $鈭� \frac{1}{x^{2} 鈭� 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $鈭� \frac{1}{\sqrt{x^{2} 鈭� 4}} d x$ .

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving $鈭� \frac{1}{x^{2} + 4} d x .$

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $鈭� {(x^{2} + 4)}^{鈭� 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} 鈭� a^{2}}$ .

(f) True or False: Trigonometric substitution doesn鈥檛 solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

(h) True or False: When using trigonometric substitution with $x = a s e c u$ , we must consider the cases $x > a$ and $x < - a$ separately.

Find three integrals in Exercises 21鈥�70 that we can anti-differentiate immediately after algebraic simplification.

Solve the integral: $鈭� \ln (3 x) d x$

See all solutions

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